Question

Se știe că :
$x - \frac{1}{x} = 3$
Atunci , calculați :
a)
$x {}^{2} + \frac{1}{x {}^{2} }$
b)
$x {}^{3} - \frac{1}{x {}^{3} }$
c)
$x {}^{4} + \frac{1}{x {}^{4} }$
Mulțumesc ! ​

Answer 1

a)

$\displaystyle x-\frac1x=3\Rightarrow \left(x-\frac1x\right)^2=3^2\Rightarrow x^2-2\cdot x\cdot\frac1x+\frac1{x^2}=9\Rightarrow x^2+\frac{1}{x^2}=11$

b)

$a^3-b^3=(a-b)(a^2+ab+b^2)$

$\displaystyle x^3-\frac{1}{x^3}=x^3-\left(\frac1{x}\right)^3=\left(x-\frac1x\right)\left(x^2+x\cdot\frac1x+\frac1{x^2}\right)=3(11+1)=36$

c)

$\displaystyle x^2+\frac1{x^2}=11\Rightarrow \left(x^2+\frac1{x^2}\right)^2=11^2\Rightarrow x^4+2\cdot x^2\cdot\frac{1}{x^2}+\frac{1}{x^4}=121\Rightarrow x^4+\frac{1}{x^4}=121-2=119$